University of Arkansas, Fayetteville
In 1998, mathematics was suddenly in the news. Thomas Hales of the University of Pittsburgh, Pennsylvania, had proved the Kepler conjecture, showing that the way greengrocers stack oranges is the most efficient way to pack spheres. A problem that had been open since 1611 was finally solved! On the television a greengrocer said: “I think that it’s a waste of time and taxpayers’ money.” I have been mentally arguing with that greengrocer ever since: today the mathematics of sphere packing enables modern communication, being at the heart of the study of channel coding and error-correction codes.
In 1611, Johannes Kepler suggested that the greengrocer’s stacking was the most efficient, but he was not able to give a proof. It turned out to be a very difficult problem. Even the simpler question of the best way to pack circles was only proved in 1940 by László Fejes Tóth. Also in the seventeenth century, Isaac Newton and David Gregory argued over the kissing problem: how many spheres can touch a given sphere with no overlaps? In two dimensions it is easy to prove that the answer is 6. Newton thought that 12 was the maximum in 3 dimensions. It is, but only in 1953 did Kurt Schütte and Bartel van der Waerden give a proof.
The kissing number in 4 dimensions was proved to be 24 by Oleg Musin in 2003. In 5 dimensions we can say only that it lies between 40 and 44. Yet we do know that the answer in 8 dimensions is 240, proved back in 1979 by Andrew Odlyzko of the University of Minnesota, Minneapolis, and Neil Sloane. The same paper had an even stranger result: the answer in 24 dimensions is 196,560. These proofs are simpler than the result for three dimensions, and relate to two incredibly dense packings of spheres, called the E8 lattice in 8-dimensions and the Leech lattice in 24 dimensions.
This is all quite magical, but is it useful? In the 1960s an engineer called Gordon Lang believed so. Lang was designing the systems for modems and was busy harvesting all the mathematics he could find.
He needed to send a signal over a noisy channel, such as a phone line. The natural way is to choose a collection of tones for signals. But the sound received may not be the same as the one sent. To solve this, he described the sounds by a list of numbers. It was then simple to find which of the signals that might have been sent was closest to the signal received. The signals can then be considered as spheres, with wiggle room for noise. To maximize the information that can be sent, these ‘spheres’ must be packed as tightly as possible.
In the 1970s, Lang developed a modem with 8-dimensional signals, using E8 packing. This helped to open up the Internet, as data could be sent over the phone, instead of relying on specifically designed cables. Not everyone was thrilled. Donald Coxeter, who had helped Lang understand the mathematics, said he was “appalled that his beautiful theories had been sullied in this way”.